Completing The Square Examples And Worksheet
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1
2Completing the Square
Completing the square is a technique for re-formatting certain algebraic expressions.
In particular, it is useful for taking quadratic expressions like
2
+ bx + c
ax
and rewriting them as
2
2
+ bx + c = a(x
+ k.
ax
h)
There are several reasons for doing this. One reason is that it allows us to easily see what the
2
vertex of the curve y = ax
+ bx + c is: it is the point (h, k).
The general method of completing the square can be shown like this:
2
y = ax
+ bx + c
b
2
= a x
+
x + c
a
2
2
b
b
= a
x +
+ c
2a
2a
2
2
b
b
= a x +
+ c
2a
4a
2
2
b
b
= a x
+ c.
2a
4a
This shows that the x-coordinate of the vertex is
b
h =
.
2a
Let’s do some examples.
2
1. Suppose y = 2x
4x
5.
2
2
y = 2(x
2x)
5
Factor the coefficient of x
out of the first two terms.
2
y = 2((x
1)
1)
5
Write the terms in the parentheses as a perfect square minus a constant.
2
y = 2(x
1)
2
5
Distribute the coefficient that you factored out in the first step.
2
y = 2(x
1)
7
Simplify and you’re done.
2
2. Suppose y = x
+ 6x
8.
2
2
2
Then y = (x + 3)
3
8 = (x + 3)
17.
This shows that the vertex of this parabola is the point ( 3, 17).
2
3. Suppose y = 3x
12x + 1.
Then
2
y = 3(x
4x) + 1
2
= 3 (x
2)
4 + 1
2
= 3(x
2)
12 + 1
2
= 3(x
2)
11.
This shows that the vertex of this parabola is the point (2, 11).
2
1
4. Suppose y =
4x
+ 7x
.
2
Then
2
7
1
y =
4 x
x
4
2
2
2
7
7
1
=
4
x
8
8
2
2
7
49
1
=
4 x
+ (4)
8
64
2
2
7
41
=
4 x
+
.
8
16
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